How would one paint a narrow circle around the edge of a circular object in blender
In general, it doesnt sound as useful to draw narrow circle in Vertex Paint. For such result you would need a lot of vertices or very specific vertices for this purpose. It would be better if you can closer describe what is your goal. Maybe you dont need to use a Vertex Paint for it.

Narrow circle you can achive by many ways ...
version 01 Material

create specific geometry and assign one material to object and create another one assigned to specific faces.


version 02 Texture

create material with Image texture

version 03 Vertex Paint

I dont know if there is a specific tool for it (Sculpt mode had some paint from curve). One way that lays on vertices position as well - you can create a Vertex Group by assigning vertices to a group ... (here displayed in Weight Paint mode)


... switch to Vertex Paint mode and use Vertex Color from Weight operator.

version 04 Paint Editor

in Image editor in 2.91 go to side panel (N) Tool - Stroke - Stroke Method - Curve, add a new and with Ctrl manually create a circle shape from curve, than press Draw